Optimal. Leaf size=74 \[ \frac{1}{4} i \text{PolyLog}\left (2,e^{2 i x}\right )+\frac{i x^2}{4}+\frac{x}{4}-\frac{1}{2} x \log \left (1-e^{2 i x}\right )+\frac{1}{2} x \log (\sin (x))+\frac{1}{4} \sin (x) \cos (x)-\frac{1}{2} \sin (x) \cos (x) \log (\sin (x)) \]
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Rubi [A] time = 0.109389, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.125, Rules used = {2635, 8, 2554, 12, 6742, 3717, 2190, 2279, 2391} \[ \frac{1}{4} i \text{PolyLog}\left (2,e^{2 i x}\right )+\frac{i x^2}{4}+\frac{x}{4}-\frac{1}{2} x \log \left (1-e^{2 i x}\right )+\frac{1}{2} x \log (\sin (x))+\frac{1}{4} \sin (x) \cos (x)-\frac{1}{2} \sin (x) \cos (x) \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rule 2554
Rule 12
Rule 6742
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log (\sin (x)) \sin ^2(x) \, dx &=\frac{1}{2} x \log (\sin (x))-\frac{1}{2} \cos (x) \log (\sin (x)) \sin (x)-\int \frac{1}{2} \cot (x) (x-\cos (x) \sin (x)) \, dx\\ &=\frac{1}{2} x \log (\sin (x))-\frac{1}{2} \cos (x) \log (\sin (x)) \sin (x)-\frac{1}{2} \int \cot (x) (x-\cos (x) \sin (x)) \, dx\\ &=\frac{1}{2} x \log (\sin (x))-\frac{1}{2} \cos (x) \log (\sin (x)) \sin (x)-\frac{1}{2} \int \left (-\cos ^2(x)+x \cot (x)\right ) \, dx\\ &=\frac{1}{2} x \log (\sin (x))-\frac{1}{2} \cos (x) \log (\sin (x)) \sin (x)+\frac{1}{2} \int \cos ^2(x) \, dx-\frac{1}{2} \int x \cot (x) \, dx\\ &=\frac{i x^2}{4}+\frac{1}{2} x \log (\sin (x))+\frac{1}{4} \cos (x) \sin (x)-\frac{1}{2} \cos (x) \log (\sin (x)) \sin (x)+i \int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx+\frac{\int 1 \, dx}{4}\\ &=\frac{x}{4}+\frac{i x^2}{4}-\frac{1}{2} x \log \left (1-e^{2 i x}\right )+\frac{1}{2} x \log (\sin (x))+\frac{1}{4} \cos (x) \sin (x)-\frac{1}{2} \cos (x) \log (\sin (x)) \sin (x)+\frac{1}{2} \int \log \left (1-e^{2 i x}\right ) \, dx\\ &=\frac{x}{4}+\frac{i x^2}{4}-\frac{1}{2} x \log \left (1-e^{2 i x}\right )+\frac{1}{2} x \log (\sin (x))+\frac{1}{4} \cos (x) \sin (x)-\frac{1}{2} \cos (x) \log (\sin (x)) \sin (x)-\frac{1}{4} i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\frac{x}{4}+\frac{i x^2}{4}-\frac{1}{2} x \log \left (1-e^{2 i x}\right )+\frac{1}{2} x \log (\sin (x))+\frac{1}{4} i \text{Li}_2\left (e^{2 i x}\right )+\frac{1}{4} \cos (x) \sin (x)-\frac{1}{2} \cos (x) \log (\sin (x)) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0555071, size = 59, normalized size = 0.8 \[ \frac{1}{8} \left (2 i \text{PolyLog}\left (2,e^{2 i x}\right )+2 x \left (i x-2 \log \left (1-e^{2 i x}\right )+2 \log (\sin (x))+1\right )+\sin (2 x) (1-2 \log (\sin (x)))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 146, normalized size = 2. \begin{align*}{\frac{i}{8}}\ln \left ( 2\,\sin \left ( x \right ) \right ){{\rm e}^{2\,ix}}-{\frac{i}{16}}{{\rm e}^{2\,ix}}-{\frac{i}{2}}\ln \left ({{\rm e}^{ix}} \right ) \ln \left ( 2\,\sin \left ( x \right ) \right ) -{\frac{i}{4}} \left ( \ln \left ({{\rm e}^{ix}} \right ) \right ) ^{2}+{\frac{i}{2}}\ln \left ({{\rm e}^{ix}} \right ) \ln \left ({{\rm e}^{ix}}+1 \right ) +{\frac{i}{2}}{\it dilog} \left ({{\rm e}^{ix}}+1 \right ) -{\frac{i}{2}}{\it dilog} \left ({{\rm e}^{ix}} \right ) -{\frac{i}{8}}{{\rm e}^{-2\,ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) +{\frac{i}{16}}{{\rm e}^{-2\,ix}}-{\frac{i}{4}}\ln \left ({{\rm e}^{ix}} \right ) -{\frac{i}{8}}\ln \left ( 2 \right ){{\rm e}^{2\,ix}}+{\frac{i}{8}}\ln \left ( 2 \right ){{\rm e}^{-2\,ix}}+{\frac{i}{2}}\ln \left ( 2 \right ) \ln \left ({{\rm e}^{ix}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.38052, size = 140, normalized size = 1.89 \begin{align*} \frac{1}{4} i \, x^{2} - \frac{1}{2} i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + \frac{1}{2} i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) - \frac{1}{4} \, x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{4} \, x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{4} \,{\left (2 \, x - \sin \left (2 \, x\right )\right )} \log \left (\sin \left (x\right )\right ) + \frac{1}{4} \, x + \frac{1}{2} i \,{\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + \frac{1}{2} i \,{\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.62214, size = 462, normalized size = 6.24 \begin{align*} -\frac{1}{4} \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac{1}{2} \,{\left (\cos \left (x\right ) \sin \left (x\right ) - x\right )} \log \left (\sin \left (x\right )\right ) + \frac{1}{4} \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{4} \, x + \frac{1}{4} i \,{\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac{1}{4} i \,{\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac{1}{4} i \,{\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac{1}{4} i \,{\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (\sin{\left (x \right )} \right )} \sin ^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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